TACL 2013:Papers with Abstracts

Abstract. Classically, a Tychonoff space is called strongly 0-dimensional if its Stone-Cech compactification is 0-dimensional, and given the familiar relationship between spaces and frames it is then natural to call a completely regular frame strongly 0-dimensional if its compact completely regular coreflection is 0-dimensional (meaning: is generated by its complemented elements). Indeed, it is then seen immediately that a Tychonoff space is strongly 0-dimensional iff the frame of its open sets is strongly 0-dimensional in the present sense. This talk will provide an account of various aspects of this notion.

Abstract. The method of canonical formulas is a powerful tool for investigating intuitionistic and modal logics. In this talk I will discuss an algebraic approach to this method. I will mostly concentrate on the case of intuitionistic logic. But I will also review the case of modal logic and possible generalizations to substructural logic.

Abstract. The analysis of coproducts in varieties of algebras has generally been variety-specific, relying on tools tailored to particular classes of algebras. A recurring theme, however, is the use of a categorical duality. Among the dualities and topological representations in the literature, natural dualities are particularly well behaved with respect to coproduct. Since (multisorted) natural dualities are based on hom-functors, they send coproducts into cartesian products.We carry out a systematic study of coproducts for finitely generated quasivarieties A that admit a (term) reduct in the variety D of bounded distributive lattices.

Abstract. (This is joint work with Nick Bezhanishvili).In the first part of our contribution, we review and compare existing constructions of finitely generated free algebras in modal logic focusing on step-by-step methods. We discuss the notions of step algebras and step frames arising from recent investigations, as well as the role played by finite duality.

In the second part of the contribution, we exploit the potential of step frames for investigating proof-theoretic aspects. This includes developing a method which detects when a specific rule-based calculus Ax axiomatizing a given logic L has the so-called bounded proof property. This property is a kind of an analytic subformula property limiting the proof search space. We prove that every finite conservative step frame for Ax is a p-morphic image of a finite Kripke frame for L iff Ax has the bounded proof property and L has the finite model property. This result, combined with a `step version' of the classical correspondence theory turns out to be quite powerful in applications.

Abstract. Proof theory can provide useful tools for tackling problems in algebra. In particular, Gentzen systems admitting cut-elimination have been used to establish decidability, complexity, amalgamation, admissibility, and generation results for varieties of residuated lattices corresponding to substructural logics. However, for classes of algebras bearing some family resemblance to groups, such as lattice-ordered groups, MV-algebras, BL-algebras, and cancellative residuated lattices, the proof-theoretic approach has met so far only with limited success.

The main aim of this talk will be to introduce proof-theoretic methods for the class of lattice-ordered groups and to explain how these methods can be used to obtain new syntactic proofs of two core theorems: namely, Holland's result that this class is generated as a variety by the lattice-ordered group of order-preserving automorphisms of the real numbers, and the decidability of the word problem for free lattice-ordered groups.

Abstract. In this talk I shall discuss a general category-theoretic structure for modelling conditional independence. The standard notion of conditional independence in probability theory provides a motivating example. But other rather different examples arise in many contexts: computability theory, nominal sets (used to model `names' in computer science), separation logic (used to reason about heap memory in computer science), and others.

Category-theoretic structure common to these examples can be axiomatized by the notion of a category with local independent products, which combines fibrational and symmetric monoidal structure in a somewhat particular way. In the talk I shall expound this notion, and I shall present several illustrative examples of such structure. If time permits, I may also describe some curious connections with topos theory.

Abstract. My research concerns a construction of "logical schemes," geometric entitieswhich represent logical theories in much the same way that algebraic schemesrepresent rings. These involve two components: a semantic spectral spaceand a syntactic structure sheaf. As in the algebraic case, we can recover atheory from its scheme representation (up to a conservative completion) andthe structure sheaf is local in a certain logical sense. From these ane pieceswe can build up a 2-category of logical schemes which share some of the niceproperties of algebraic schemes.

Abstract. Topos-theoretic semantics for modal logic usually uses structures induced by a surjective geometric morphism between toposes. This talk develops an algebraic generalization of this framework. We take internal adjoints between certain internal frames within a topos, which provides semantics for (intuitionistic) higher-oder modal logic.

Abstract. In analysis, truncation is the operation of replacing a nonnegative real-valued function a (x) by its pointwise meet a (x) ∧ 1 with the constant $1$ function. A vector lattice A is said to be closed under truncation if a ∧ 1 ∈ A for all a ∈ A+. Note that A need notcontain 1 itself.

Truncation is fundamental to analysis. To give only one example, Lebesgue integration generalizes beautifully to any vector lattice of real-valued functions on a set X, provided the vector lattice is closed under truncation. But vector lattices lacking this property may have integrals which cannot be represented by any measure on X. Nevertheless, when the integral is formulated in a context broader than RX, for example in pointfree analysis, the question oftruncation inevitably arises.

What is truncation, or more properly, what are its essential properties? In this paper we answer this question by providing the appropriate axiomatization, and then go on to present several representation theorems. The first is adirect generalization of the classical Yosida representation of an archimedean vector lattice with order unit. The second is a direct generalization of Madden's pointfree representation of archimedean vector lattices. If time permits, we briefly discuss a third sheaf representation which has no direct antecedent in the literature.

However, in all three representations the lack of a unit forces a crucial distinction from the corresponding unital representation theorem. The universal object in each case is some sort of family of continuous real-valued functions. The difference is that these functions must vanish at a specified point of the underlying space or locale or sheaf space. With that adjustment, the generalization from units to truncations goes remarkably smoothly.

Abstract. It was left as an open problem in (Bezhanishvili and Gabelaia, 2011) whether a connected normal extension of S4 without FMP is also the modal logic of some subalgebra of R+.Our purpose here is to solve this problem affirmatively by showing that each connected normal extension of S4 (with or without FMP) is in fact the modal logic of some subalgebra of R+. We also prove that each normal extension of S4 (with or without FMP) is the modal logic of a subalgebra of Q+, as well as the modal logic of a subalgebra of C+.

Abstract. For a commutative ring R, we introduce the notion of a Specker R-algebra and show that Specker R-algebras are Boolean powers of R. For an indecomposable ring R, this yields an equivalence between the category of Specker R-algebras and the category of Boolean algebras. Together with Stone duality this produces a dual equivalence between the category of Specker R-algebras and the category of Stone spaces.

Abstract. The main aim of this talk is twofold. Firstly, to present an elementarymethod based on Farkas' lemma for rationals how to embed any finite partial subalgebraof a linearly ordered MV-algebra into Q \ [0; 1] and then to establish a new elementaryproof of the completeness of the Lukasiewicz axioms for which the MV-algebras communityhas been looking for a long time. Secondly, to present a direct proof of Di Nola'srepresentation Theorem for MV-algebras and to extend his results to the restriction ofthe standard MV-algebra on rational numbers.

Abstract. We consider the knotted structural rule x<sup>m</sup>≤x<sup>n</sup> for n different than m and m greater or equal than 1. Previously van Alten proved that commutative residuated lattices that satisfy the knotted rule have the finite embeddability property (FEP). Namely, every finite partial subalgebra of an algebra in the class can be embedded into a finite full algebra in the class. In our work we replace the commutativity property by some slightly weaker conditions. Particularly, we prove the FEP for the variety of residuated lattices that satisfy the equation xyx=x<sup>2</sup>y and the knotted rule. Furthermore, we investigate some generalizations of this noncommutative property by working with equations that allow us to move variables. We also note that the FEP implies the finite model property. Hence the logics modeled by these residuated lattices are decidable.

Abstract. The idea of two-layer modal logics is inspired by the treatment of probability inside mathematical fuzzy logic, pioneered by Hajek and recently studied by numerous authors in numerous papers. Such logics are used in order to deal with a certain property of formulas of the base logic using a suitable `upper' logic (the seminal example being the probability of classical events formalized inside Lukasiewicz logic). The primary aim of this paper is to provide a new general framework for two-layer modal logics that encompasses the current state of the art and paves the way for future development. Diverting for the area of mathematical fuzzy logic, we show how one can construct such modal logic over an arbitrary non-classical logic (under certain technical requirements) with a modality interpreted by an arbitrary measure. We equip the resulting logics with a semantics of measured Kripke frames and prove corresponding completeness theorems. As an illustration of our results, we reprove Hajek's completeness result for Fuzzy Probability logic over Lukasiewicz logic.

Abstract. Grounding on defining relations of a finitely presentable subdirectly irreducible (s.i.) algebra in a variety with a ternary deductive term (TD), we define the characteristic identity of this algebra. For finite s.i. algebras the characteristic identity is equivalent to the identity obtained from Jankov formula. In contrast to Jankov formula, characteristic identity is relative to a variety and even in the varieties of Heyting algebras there are the characteristic identities not related to Jankov formula. Every subvariety of a given locally finite variety with a TD term admits an optimal axiomatization consisting of characteristic identities. There is an algorithm that reduces any finite system of axioms of such a variety to an optimal one. Each variety with a TD term can be axiomatized by characteristic identities of partial algebras, and in certain cases these identities are related to the canonical formulas.

Abstract. The algorithmic correspondence theory is extended to mu-calculi with a non-classical base. We focus in particular on the language of bi-intuitionistic modal mu-calculus, and we enhance the algorithm, or calculus for correspondence, ALBA for the elimination of monadic second order variables, so as to guarantee its success over a class including the Sahlqvist mu-formulas. Key to the soundness of this enhancement are the order-theoretic properties of the algebraic interpretation of the fixed point operators.

Abstract. Sahlqvist-style correspondence results remain a perennial theme and an active topic of research within modal logic. Recently there has been interest in extending classical results in this area to the modal mu-calculus. We show how the `calculus of correspondence' and the ALBA algorithm (Conradie and Palmigiano, 2012) can be extended to the intuitionistic mu-calculus, and be used to derive FO+LFP frame correspondents for formulas of that logic. We define the class of recursive mu-inequalities, which we compare it with related classes in the literature including the Sahlqvist mu-formulas of van Benthem, Bezhanishvili and Hodkinson. We show that the ALBA algorithm succeeds in reducing every recursive mu-inequality, and hence that every recursive mu-inequality has a frame correspondent in FO+LFP.

Abstract. The notion of structural completeness has received considerable attention for many years. A translating to algebra gives: a quasivariety is structurally complete if it is generated by its free algebras. It appears that many deductive systems (quasivarieties), like S5 or MV<sub>n</sub> fails structural completeness for a rather immaterial reason. Therefore the adjusted notion was introduced: almost structural completeness. We investigate almost structural completeness from an algebraic perspective and obtain a characterization of this notion for quasivarieties.

Abstract. In this paper we characterize the regular medial algebras, the paramedial n-ary groupoids with a regular element, the paramedial algebras with a regular element and the regular paramedial algebras. Also, we characterize paramedial, co-medial and co-paramedial pairs of quasigroup operations and paramedial, co-medial and co-paramedial algebras with the quasigroup operations.

Abstract. We present the results of our research on stone-type dualities for certain classes of ordered algebras that do not fall within the scope of extended Priestley-duality. In a forthcoming paper we study a new spectral-like duality for the class of distributive Hilbert algebras with infimum. We explain the main facts of that duality and we outline how the same strategy could be used for getting a Priestley-style duality for the same class of algebras, as well as dualities for other classes of algebras.

Abstract. One of the authors introduced in [1] a calculus ofcircular proofs for studying the computability arising from thefollowing categorical operations: finite products and coproducts,initial algebras, final coalgebras. The calculus of[1] is cut-free; yet, even if sound and complete forprovability, it lacks an important property for the semantics ofproofs, namely fullness w.r.t. the class of natural categorical modelscalled μ-bicomplete category in [2].

We fix, with this work, this problem by adding the cut rule to thecalculus. To this goal, we need to modifying the syntacticalconstraints on the cycles of proofs so to ensure soundness of thecalculus and at same time local termination of cut-elimination. Theenhanced proof system fully represents arrows of the intended model, afree μ-bicomplete category. We also describe a cut-eliminationprocedure as a model of computation arising from the above mentionedcategorical operations. The procedure constructs a cut-freeproof-tree with infinite branches out of a finite circular proof withcuts.

Abstract. We obtain representation theorems for residuated lattices. The representing structure consists of special self maps on an ordered set. We prove two types of theorems; one that generalizes Cayley's theorem for groups/monoids and one (for special residuated lattices) that generalizes Holland's theorem for lattice-ordered groups. Our results are presented in the language of idempotent semirings and semimodules, and they are first established for these types of structures.

Abstract. The existence of lateral completions of ℓ-groups is an old problem that was first solved, for conditionally complete vector lattices, by Nakano. The existence and uniqueness of lateral completions of representable ℓ-groups was first obtained as a consequence of the orthocompletions of Bernau, and later the proofs were simplified by Conrad, who also proved the existence and uniqueness of lateral completions of ℓ-groups with zero radical. Finally, Bernau solved the problem for ℓ-groups in general.

In this work, we address the problem of the existence and uniqueness of lateral, projectable, and strongly projectable completions of residuated lattices. In particular, we push the methods of Conrad through to the case of the representable GMV-algebras.

The leading idea is to construct, for any given semilinear residuated lattice, an orthocomplete extension such that the former is dense in the latter. This extension is obtained as the direct limit of a family of residuated lattices that are constructed using maximal partitions of the algebra of polars of the original residuated lattice.

In order to complete the proof we still need another hypothesis, which is an abstraction of the condition of double negation in which commutativity and integrality have been dropped, and determines the wide class of Generalized MV-algebras. This, together with the density, allows us to obtain the completions of the given residuated lattice.

Abstract. Skura syntactically characterised intuitionistic propositional logic among all intermediate logics by means of a Łukasiewicz-style refutation system. Another such syntactic characterisation is given by Iemhoff in terms of admissible rules. Here we offer a bridge between these results. That is to say, we provide sufficient conditions under which admissible rules yield a refutation system fully characterising the logic. In particular, we give a characterisation of the Gabbay–de Jongh logics by means of refutation systems employing ideas from admissibility.

Abstract. We develop a family of display-style, cut-free sequent calculi for dynamic epistemic logics on both an intuitionistic and a classical base. Like the standard display calculi, these calculi are modular: just by modifying the structural rules according to Dosen’s principle, these calculi are generalizable both to different Dynamic Logics (Epistemic, Deontic, etc.) and to different propositional bases (Linear, Relevant, etc.). Moreover, the rules they feature agree with the standard relational semantics for dynamic epistemic logics.

Abstract. A description of finitely generated free monadic MV-algebras anda characterization of projective monadic MV-algebras in locally finitevarieties is given. It is shown that unification type of locally finitevarieties is unitary.

Abstract. A quite general order-theoretical approach to implicative structures leads to consider implicative groupoids, which form a wide class of algebras including residuated lattices and their reasonable generalizations. Implicative groupoids find out to be special instances of suitable relational systems and are objects of categories, semicategories and precategories whose morphisms are generalizations of Galois connections.

Abstract. This paper adds monotonicity and antitonicity information to the typed lambda calculus, thereby providing a foundation for the Monotonicity Calculus first developed by van Benthem and others. We establish properties of the type system, propose a syntax, semantics, and proof calculus, and prove completeness for the calculus with respect to hierarchies of monotone and antitone functions over base preorders.

Abstract. In [Riečanová Z, Zajac M.: Hilbert Space Effect-Representations of Effect Algebras] it was shown that an effect algebra E with an ordering set M of states can by embedded into a Hilbert space effect algebra E(l<sub>2</sub>(M)). We consider the problem when its effect algebraic MacNeille completion Ê can be also embedded into the same Hilbert space effect algebra E(l<sub>2</sub>(M)). That is when the ordering set M of states on E can be be extended to an ordering set of states on Ê. We give an answer for all Archimedean MV-effect algebras and Archimedean atomic lattice effect algebras.

Abstract. N4-lattices are the algebraic semantics of paraconsistent Nelson logic, which was introduced as an inconsistency-tolerant counterpart of the better-known logic of Nelson. Paraconsistent Nelson logic combines interesting features of intuitionistic, classical and many-valued logics (e.g., Belnap-Dunn four-valued logic); recent work has shown that it can also be seen as one member of the wide family of substructural logics.The work we present here is a contribution towards a better topological understanding of the algebraic counterpart of paraconsistent Nelson logic, namely a variety of involutive lattices called N4-lattices.

Abstract. The category of effect algebras is the Eilenberg-Moore category for the monad arising from the free-forgetful adjunction between categories of bounded posetsand orthomodular posets.

In the category of effect algebras, an observable is a morphism whose domainis a Boolean algebra. The characterization of subsets of ranges of observables isan open problem.For an interval effect algebra E, a witness pair for a subset of S is an objectliving within E that "witnesses existence" of an observable whose rangeincludes S. We prove that there is an adjunction between the poset of allwitness pairs of E and the category of all partially inverted E-valuedobservables.

Abstract. We show how to implement an effective decision procedure to check if a propositional Basic Logic formula is a tautology. For a formula with $n$ variables, the procedure consists of a translation, depending on $n$, from Basic Logic to the language of Satisfiability Modulo Theories SMT-LIB2 using the theory of quantifier free linear real arithmetic. Many efficient SMT-solvers exist to decide formulas in the SMT-LIB2 language. We also study finitely generated varieties of Basic Logic (BL-)algebras and give a description of the lattice of these varieties. Extensions to finitely generated varieties of Generalized BL-algebras are discussed, and a simple connection between finite GBL-algebras and finite closure algebras is noted.

Abstract. Along the lines of recent investigations combining many-valued and modal systems, we address the problem of defining and axiomatizing the least modal logic over the four-element Belnap lattice. By this we mean the logic determined by the class of all Kripke frames where the accessibility relation as well as semantic valuations are four-valued. Our main result is the introduction of two Hilbert-style calculi that provide complete axiomatizations for, respectively, the local and the global consequence relations associated to the class of all four-valued Kripke models. Our completeness proofs make an extensive and profitable use of algebraic and topological techniques; in fact, our algebraic and topological analyses of the logic have, in our opinion, an independent interest and contribute to the appeal of our approach.

Abstract. In 2007, Maddux observed that certain classes of representable relationalgebras (RRAs) form sound semantics for some relevant logics. In particular,(a) RRAs of transitive relations are sound for R, and (b) RRAs of transitive,dense relations are sound for RM. He asked whether they were complete aswell. Later that year I proved a modest positive result in a similar direction,namely that weakly associative relation algebras, a class (much) larger thanRRA, is sound and complete for positive relevant logic B. In 2008, Mikulas proved anegative result: that RRAs of transitive relations are not complete for R. His proof is indirect: he shows that the quasivariety of appropriatereducts of transitive RRAs is not finitely based. Later Maddux re-establishedthe result in a more direct way. In 2010, Maddux proved a contrasting positive result: that transitive, dense RRAs are complete for RM. He found an embedding of Sugihara algebras into transitive, dense RRAs. I will show that if we give up the requirement of representability, the positive result holds for R as well. To be precise, the following theorem holds.

Theorem. Every normal subdirectly irreducible De Morgan monoid in the languagewithout Ackermann constant can be embedded into a square-increasing relationalgebra. Therefore, the variety of such algebras is sound and complete for R.

Abstract. Moebius transform is introduced for the representation of functions over a semisimple MV-algebra. The class of such functions is singled out and the properties of the generalized Moebius transform are studied. The main tool is the Vietoris space of all closed subsets of the maximal spectral space and the associated MV-algebra of continuous functions.

Abstract. Finite distributive lattices with antitone involutions (= basic algebras) are studied; it is proved that their underlying lattices are isomorphic to direct products of finite chains, and hence finite distributive basic algebras can be constructed by “perturbing” finite MV-algebras, and moreover, under certain natural conditions, they even coincide with finite MV-algebras.

Abstract. Using Vaggione's concept of central element in a double pointed algebra, weintroduce the notion of Boolean like variety as a generalisation ofBoolean algebras to an arbitrary similarity type. Appropriately relaxing therequirement that every element be central in any member of the variety, weobtain the more general class of semi-Boolean like varieties, whichstill retain many of the pleasing properties of Boolean algebras. We provethat a double pointed variety is discriminator iff it is semi-Boolean like,idempotent, and 0-regular. This theorem yields a new Maltsev-stylecharacterisation of double pointed discriminator varieties.Moreover, we point out the exact relationship between semi-Boolean-likevarieties and the quasi-discriminator varieties, and we providesemi-Boolean-like algebras with an explicit weak Boolean productrepresentation with directly indecomposable factors. Finally, we discuss idempotentsemi-Boolean-like algebras. We consider a noncommutative generalisation of Boolean algebras and prove - along the lines of similar results available for pointed discriminator varieties or for varieties with a commutative ternary deduction term that every idempotent semi-Boolean-like variety is term equivalent to a variety of noncommutative Boolean algebras with additional operations.

Abstract. Walker's cancellation theorem says that if B + Z isisomorphic to C + Z in the category of abeliangroups, then B is isomorphic to C. We construct an example ina diagram category of abelian groups where the theorem fails. As aconsequence, the original theorem does not have a constructiveproof. In fact, in our example B and C are subgroups ofZ<sup>2</sup>. Both of these results contrast with a group whoseendomorphism ring has stable range one, which allows aconstructive proof of cancellation and also a proof in any diagramcategory.

Abstract. This is the first part of a series of two abstract, the second one being by Daniel McNeill.

If X is any topological space, its collection of opens sets O(X) is a complete distributive lattice and also a Heyting algebra. When X is equipped with a distinguished basis D(X) for its topology, closed under finite meets and joins, one can investigate situations where D(X) is also a Heyting subalgebra of O(X).

Recall that X is a spectral space if it is compact and T0, its collection D(X) of compact open subsets forms a basis which is closed under finite intersections and unions, and X is sober. By Stone duality, spectral spaces are precisely the spaces arising as sets of prime ideals of some distributive lattice, topologised with the Stone or hull-kernel topology. Specifically, given such a spectral space X, its collection of compact open sets D(X) is (naturally isomorphic to) the distributive lattice dual to X under Stone duality. We are going to exhibit a significant class of such spaces for which D(X) is a Heyting subalgebra of O(X).

We work with lattice-ordered Abelian groups and vector spaces. Using Mundici’s Gamma-functor the results can be rephrased in terms of MV-algebras, the algebraic semantics of Lukasiewicz infinite-valued propositional logic.

Let (G,u) be a finitely presented vector lattice (or Q-vector lattice, or l-group) G equipped with a distinguished strong order unit u. It turns out that Spec(G,u), i.e. the the space of prime ideals of (G,u) topologised with the hull-kernel topology, is a compact spectral space. Our first main result states that the collection D(Spec(G,u)) of compact open subsets of Spec(G,u) is a Heyting subalgebra of the Heyting algebra of open subsets O(Spec(G,u)).

As a consequence, we also prove that the subspace Min(G,u) of minimal prime ideals of G is a Boolean space, i.e. a compact Hausdorff space whose clopen sets form a basis for the topology.

Further, for any fixed maximal ideal m of G, the set l(m) of prime ideals of G contained in m, equipped with the subspace topology, is a spectral space, and the subspace Min(l(m)) of l(m) is a Boolean space.

Abstract. This is the second part of a series of two abstracts, the first being by Andrea Pedrini. For background and notation on lattice-ordered Abelian groups, vector lattices and Q-vector lattices, and their spectral spaces, please see her submission.We consider the tools of Stone duality and the absolute applied to lattice-ordered Abelian groups, vector lattices and Q-vector lattices. Given a lattice-ordered Abelian group or Q-vector lattice, G, this leads to an interesting parallel between Min(G) and the absolute of Max(G).

Abstract. This paper is a contribution to the presentation of fractal sets in terms of final coalgebras.The first result on this topic was Freyd's Theorem: the unit interval [0,1] is the final coalgebra ofa certain functor on the category of bipointed sets. Leinster 2011 offers a sweeping generalization of this result. He is able to represent many of what would be intuitively called "self-similar" spaces using (a) bimodules (also called profunctors or distributors),(b) an examination of non-degeneracy conditions on functors of various sorts; (c) a construction offinal coalgebras for the types of functors of interest using a notion of resolution. In addition to the characterization of fractals sets as sets, his seminal paper also characterizes them as topological spaces.

Our major contribution is to suggest that in many cases of interest, point (c) above on resolutions is not needed in the construction of final coalgebras. Instead, one may obtain a number of spaces ofinterest as the Cauchy completion of an initial algebra,and this initial algebra is the set of points in a colimit of an omega-sequence of finite metric spaces.This generalizes Hutchinson's 1981 characterization of fractal attractors as closures of the orbits of the critical points. In addition to simplifying the overall machinery, it also presents a metric space which is ``computationally related'' to the overall fractal. For example, when applied to Freyd's construction, our method yields the metric space.of dyadic rational numbers in [0,1].

Our second contribution is not completed at this time, but it is a set of results on \emph{metric space} characterizations of final coalgebras. This point was raised as an open issue in Hasuo, Jacobs, and Niqui 2010,and our interest in quotient metrics comes from their paper. So in terms of (a)--(c) above, our work develops (a) and (b) in metric settings while dropping (c).

Abstract. We consider the representation of each extension of the modal logic S4 as sum of two components. The first component in such a representation is always included in Grzegorczyk logic and hence contains "modal resources" of the logic in question, while the second one uses essentially the resources of a corresponding intermediate logic. We prove some results towards the conjecture that every S4-logic has a representation with the least component of the first kind.

Abstract. For any BL-algebra L, we construct an associated lattice ordered Abelian group that coincides with the Chang’s l-group of an MV-algebra when the BL-algebra is an MV-algebra. We prove that the Chang’s group of the MV-center of any BL-algebra L is a direct summand in the above group. We also find a direct description of the complement of the Chang’s group of the MV-center in terms of the filter of dense elements of L. Finally, we compute some examples of the introduced group.

Abstract. Several familiar results about normal and extremally disconnected (classical or pointfree) spacesshape the idea that the two notions are somehow dual to each other and can therefore be studied inparallel. In this talk we discuss the source of this ‘duality’ and show that each pair of parallelresults can be framed by the ‘same’ proof. The key tools for this purpose are relative notionsof normality, extremal disconnectedness, semicontinuity and continuity (with respect to a fixedclass of complemented sublocales of the given locale) that bring and extend to locale theory avariety of well-known classical variants of normality and upper and lower semicontinuities in ailluminating unified manner. This approach allows us to unify under a single localic proof a great variety ofclassical insertion results, as well as their corresponding extension results.

Abstract. A state operator on effect algebras is introduced as an additive, idempotent and unital mapping from the effect algebra into itself. The definition is inspired by the definition of an internal state on MV-algebras, recently introduced by Flaminio and Montagna. We study state operators on convex effect algebras, and show their relations with conditional expectations on operator algebras.

Abstract. Hybrid logic extends modal logic with a special sort of variables, called nominals, which are evaluated to singletons in Kripke models by valuations, thus acting as names for states in models. Various syntactic mechanisms for exploiting and enhancing the expressive power gained through the addition of nominals can be included, most characteristically the satisfaction operator, @_ip, allowing one to express that p holds at the world named by a nominal i.

R.A. Bull famously proved that each normal extension of S4.3 has the finite modelproperty. In the current paper, we prove a hybrid analogue of Bull's result. Like the proof of Bull's original result, ours is algebraic, and thus our secondary aim with this work is to illustrate the usefulness of algebraic methods within hybrid logic research, a field where such methods have been largely ignored.

Abstract. A new semantics with the finite model property is provided and used to establish decidability for Gödel modal logics based on (crisp or fuzzy) Kripke frames combined locally with Gödel logic. A similar methodology is also used to establish decidability, indeed co-NP-completeness, for a Gödel S5 logic that coincides with the one-variable fragment of first-order Gödel logic.

Abstract. Classically, Hopf algebras are defined on the basis of modules over commutative rings. The present study seeks to extend the Hopf algebra formalism to a more general universal-algebraic setting, entropic varieties, including (pointed) sets, barycentric algebras, semilattices, and commutative monoids. The concept of a setlike (or grouplike) element may be defined, and group algebras constructed, in any such variety. In particular, group algebras within the variety of barycentric algebras consist precisely of finitely supported probability distributions on groups. For primitive elements and group quantum doubles, the natural universal-algebraic classes are entropic Jónsson-Tarski varieties (such as semilattices or commutative monoids). There, the tensor algebra on any algebra is a bialgebra, and the set of primitive elements of a Hopf algebra forms an abelian group. Coalgebra congruences on comonoids in entropic varieties are also studied.

Abstract. We generalize the double negation construction of Boolean algebras in Heytingalgebras, to a double negation construction of the same in Visser algebras (alsoknown as basic algebras).This result allows us to generalize Glivenko's Theorem from intuitionisticpropositional logic and Heyting algebras to Visser's basic propositional logicand Visser algebras.

Abstract. We consider propositional language endowed with three families of modal operators:operators of the first type are interpreted by iterations of the Cantor derivative, operators of the second and the third types are interpreted by the image and the preimage of finitary operations on a topological space. We describe logics in this language that are sound and complete w.r.t. operations of a certain kind: injective, continuous, open, closed, continuous and discrete, closed of finite rank, separately continuous and separately discrete, etc. These results are based on our recent work "On interactions of the Cantor derivative and images of finitary maps between topological spaces" (2012, submitted to Topology and its Applications).

Abstract. We consider shifted products of modal algebras and logics first introduced by Y. Hasimoto in 2000. For logics this operation is similar to the well-known usual product but it is logically invariant. We prove the conjecture of D. Gabbay that shifted products act on Boolean algebras exactly as tensor products, so we call them tensor products of modal algebras. We also propose a filtration technique for models based on tensor products and obtain some decidability results.

Abstract. Two weakened versions of the constant domains principle D are considered. The Kripke sheaf incompleteness of superintuitionistic predicate logics, obtained by adding these weakened principles to predicate versions of superintuitionistic propositional logics of finite slices, is claimed. So a difference between these principles and the principle D is shown. The incompleteness proof uses the functor semantics.

Abstract. The concept of quantum triad has been introduced by D. Kruml, where for a given pair of quantale modules L, R over a common quantale Q, endowed with a bimorphism (a `bilinear map') to Q, a construction equipping L and R with additional module structure and another bimorphism, both compatible with the existing bimorphism and action of the quantale, was presented. As the original concept was only defined in a specific setting of categories of quantale modules, we extend it to a more universal one, which can be applied to other common algebraic structures.

Abstract. This talk provides a fuzzification procedure for topological categories, i.e., given a topological category A, there exists a topological category B, which contains A as a full concretely coreflective subcategory, and which can be considered as a fuzzification of A.

Abstract. We set up a framework that subsumes many important dualities in mathematics (Birkhoff, Stone, Priestly, Baker-Beynon, etc.) as well as the classical correspondence between polynomial ideals and affine varieties in algebraic geometry. Our main theorems provide a generalisation of Hilbert's Nullstellensatz to any (possibly infinitary) variety of algebras. The common core of the above dualities becomes then clearly visible and sets the basis to a canonical method to seek for a geometric dual to any given variety of algebras.

Abstract. A poset model of a topological space X is a poset P such that the subspace Max(P) of the Scott space ΣP consisting of all maximal points of P is homeomorphic to X. Every T<sub>1</sub> space has a (bounded complete algebraic) poset model. It is, however, not known whether every T<sub>1</sub> space has a dcpo model and whether every sober T<sub>1</sub> space has a dcpo model whose Scott topology is sober. In this paper we give a positive answer to these two problems. For each T<sub>1</sub> space X we shall construct a dcpo A that is a model of X, and prove that X is sober if and only if the Scott topology of A is sober. One useful by-product is a method that can be used to construct more non-sober dcpos.